After todays lesson you should be able to: Use numerical, algebraic, and graphical models to solve problems Translate from one model to another 1.1 Functions and Graphs Lets get motivated If you would like to predict the total expected rainfall in China over the next year, which type of model would be most helpful: Numerical (rainfall totals over the last 10 years) Graphical (scatter plot of rainfall totals vs year) Algebraic (formula for yearly rainfall totals) Explain your reasoning. Mathematical model mathematical structure that approximates phenomena for the purpose of studying or predicting their behavior. Mathematical modeling process of devising mathematical models Numerical Models Numbers (or data) are analyzed to gain insights into phenomena Examples The major league baseball standings Network of interrelated numbers that measure the global economy a) In what five-year period did the actual minimum wage increase the most? b) In what year did a worker earning the minimum wage enjoy the greatest purchasing power? c) What was the longest period during which the minimum wage did not increase? d) A worker on minimum wage in 1980 was earning nearly twice as much as a worker on minimum wage in 1970, and yet there was a great pressure to raise the minimum wage again. Why? YearMinimum Hourly Wage Purchasing Power in 2001 dollars Algebraic Models Uses formulas to relate variable quantities associated with the phenomena being studied. What advantage does an algebraic model have over a numerical model? A bakery sells a 9 x 13 cake for the same price as an 8 diameter round cake. If the round cake is twice the height of the rectangular cake, which option would you choose? Why? Finding an Algebraic Model A department store is having a sale in which everything is discounted 25% off the marked price. The discount is taken at the sales counter, and then a state sales tax of 6.5% and a local sales tax of 0.5% are added on. 1) If you only have $30, can you afford to buy a shirt marked $36.99? 2) If you have a credit card but are determined to spend no more than $100, what is the maximum total value of your marked purchases before you present them at the sales counter? Be prepared to support your answers. Graphical Models A visible representation of a numerical model or an algebraic model that gives insight into the relationship between variable quantities. Using a graphing calculator aids in solving problems using a graphical model Limitations of Graphical Models Graph of numerical or algebraic model is not visible (adjust screen window) Vertical Asymptotes (use the table menu on the calculator to view specific value of x) Part of graph is hidden (adjust screen window) US Prison population (Female) Model this growth graphically and use the graphical model to suggest an algebraic model. Solving Equations Graphically Method 1 1. Enter left-hand side as equation 1 2.Enter right-hand side as equation 2 3.Graph both equations 4.The solutions are the points of intersection Method 2 1.Set equation equal to zero. 2.Enter altered equation into calculator 3.The solutions are the x-intercepts (y = 0) Solve the equation x 2 = x algebraically and graphically (both methods). 1) Graph: y = 3/(2x 5). Is there an x-intercept? (Lets take a look at the table and what happens to the y-values as the x-values increase) 2) Solve graphically: x 3 1.1x 2 65.4x = 0 The behavior of a graphical model is sometimes hidden.